Law of Large Numbers for a Class of Superdiffusions

Pinsky (1996) [15] proved that the finite mass superdiffusion X corresponding to the semilinear operator Lu+ βu−αu exhibits local extinction if and only if λc ≤ 0, where λc := λc(L+β) is the generalized principal eigenvalue of L + β on R. For the case when λc > 0, it has been shown in Engländer and Turaev (2000) [8] that in law the superdiffusion locally behaves like exp[tλc] times a non-negative nondegenerate random variable, provided that the operator L + β − λc satisfies a certain spectral condition (‘product-criticality’), and that α and μ = X0 are ‘not too large’. In this article we will prove that the convergence in law used in the formulation in [8] can actually be replaced by convergence in probability. Furthermore, instead of R we will consider a general Euclidean domain D ⊆ R. As far as the proof of our main theorem is concerned, the heavy analytic method of [8] is replaced by a different, simpler and more probabilistic one. We introduce a space-time weighted superprocess (H-transformed superprocess) and use it in the proof along with some elementary probabilistic arguments. MSC 2000 subject classifications. 60J60, 60J80